We present a method for proving that a semigroup is finitely based and find some new sufficient conditions under which a monoid is finitely based. As an application, we find a class of finite monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm.
The method results in a short proof of the theorem of E. Lee that every monoid that satisfies xtxysy = xtyxsy and xytxsy = yxtxsy is finitely based. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of n-testable languages is finitely based if and only if n < 4.
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new …
We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid $M$ is said to preserve property $\mathcal{P}$ in direct products if, for any two $M$-acts $A$ and $B$, the direct product $A\times B$ has property $\mathcal{P}$ if and only if both $A$ and $B$ have property $\mathcal{P}$. It is proved that the monoids $M$ that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal $M$-act $M\times M$ is finitely generated (resp. fini…
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and is contained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ is non-finitely based. We establish this result by analyzing $τ$-terms for $M$ where $τ$ is certain non-trivial congruence on the free semigroup, that is, we…
Stalactic, taiga, sylvester and Baxter monoids arise from the combinatorics of tableaux by identifying words over a fixed ordered alphabet whenever they produce the same tableau via some insertion algorithm. In this paper, three sufficient conditions under which semigroups are finitely based are given. By applying these sufficient conditions, it is shown that all stalactic and taiga monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities an…
A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck--Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of $S$ to be finitely generated.
We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and stro…
A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there exists a finitely universal variety of monoids. The main objective of the present article is to exhibit the first examples of finitely universal varieties of monoids. The finite universality of these varieties is established by showing that the lattice of equivalence relations on every sufficiently large finite set is anti-isomorphic to some subinterval of …
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
Characterization of Cross varieties of $J$-trivial monoids
Institute of Natural Sciences and Mathematics, Ural Federal University（乌拉尔联邦大学自然科学与数学研究所）；Department of Mathematics, Nova Southeastern University（新南佛罗里达大学数学系）；School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.
The purpose of this paper is to introduce the theory of presentations of monoids acts. We aim to construct `nice' general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an $M$-act $A$ and a subact $B$ of $A$, on the one hand we construct presentations for $B$ and the Rees quotient $A/B$ using a presentation for $A$, and on the other hand we derive a presentation for $A$ from presentations for $B$ and $A/B$. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of fin…
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. We present a new pair of limit varieties of monoids and show that together with the five limit varieties of monoids previously discovered by Jackson, Zhang and Luo and the first-named author, there are exactly seven limit varieties of J-trivial monoids.
We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
A variety of universal algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. Until recently, only two explicit examples of limit varieties of monoids, constructed by Jackson, were known. Recently Zhang and Luo found the third example of such a variety. In our work, one more example of a limit variety of monoids is given.
Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascen…
We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to …
If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form $a<w>b$, where $a,b,w$ are elements of $M$ and $<w>$ denotes the monogenic semigroup generated by $w$. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements $a w^n b$ in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decomposition i…
We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization as quasi-geodesic monoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.
This article examines the three-way relationship between right coherency of a monoid $S$, solutions of equations over $S$-acts, and injectivity properties of $S$-acts. A monoid $S$ is right coherent if every finitely generated subact of every finitely presented (right) $S$-act itself has a finite presentation. Purity properties of an $S$-act $A$ may either be expressed in terms of solutions in $A$ of certain consistent sets of equations over $A$, or in terms of injectivity properties. For example, an $S$-act $A$ is absolutely pure (almost pure) if every finite consistent set of equations over …
Let $\mathfrak A$ be an alphabet and $W$ be a set of words in the free monoid ${\mathfrak A}^*$. Let $S(W)$ denote the Rees quotient over the ideal of ${\mathfrak A}^*$ consisting of all words that are not subwords of words in $W$. A set of words $W$ is called {\em finitely based} if the monoid $S(W)$ is finitely based. A word $\bf u$ is called 2-limited if each variable occurs in $\bf u$ at most twice. A {\em block} of a word $\bf u$ is a maximal subword of $\bf u$ that does not contain any linear variables. We say that a word $\bf u$ is {\em block-2-simple} if each block of $\bf u$ involves …
We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.
